Problem 77 - Contemporary Precalculus

Question

Suppose \(f(x)=A \ln x+B,\) where \(A\) and \(B\) are constants. If \(f(1)=10\) and \(f(e)=1,\) what are \(A\) and \(B ?\)

Step-by-Step Solution

Answer

Question: Determine the values of the constants A and B in the logarithmic function f(x) = A * ln(x) + B if f(1) = 10 and f(e) = 1. Solution: We found the values for A and B by using the given conditions and solving a system of linear equations. The values are A = -9 and B = 10.

Step 1: Plug in the given values into the function

Substitute f(1) = 10 and f(e) = 1 in the function f(x) = A * ln(x) + B. This will give us two separate equations as follows: 1) 10 = A * ln(1) + B 2) 1 = A * ln(e) + B

Step 2: Simplify the equations

In equation (1), ln(1) is 0, and in equation (2), ln(e) is 1. After substituting these values, we get: 1) 10 = A * 0 + B -> 10 = B 2) 1 = A * 1 + B -> 1 = A + B

Step 3: Solve the system of linear equations

Now that we have the simplified equations, we can solve this system of linear equations to find the values of A and B. Using equation (1), we know B = 10. Now substitute B into equation (2) to solve for A: 1 = A + 10 Subtract 10 from both sides to get: A = -9

Step 4: Write down the values of A and B

Now that we have found the values for A and B, we can write them down: A = -9 B = 10 So, the function f(x) is given by f(x) = -9 * ln(x) + 10.

Key Concepts

natural logarithm

The natural logarithm is a special type of logarithm that uses the constant \( e \) as its base. The symbol for the natural logarithm is \( \ln \). The constant \( e \) is approximately equal to 2.71828, and is an irrational number that appears frequently in mathematics, especially in calculus and complex analysis.

Natural logarithms have certain properties that make them very useful:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln(e) = 1 \) because \( e^1 = e \).
• For any positive number \( a \), \( \ln(e^a) = a \) because its exponential form translates back into \( a \).

These properties simplify the process of solving equations that involve natural logarithms, just like in the exercise where substituting \( \ln(1) \) and \( \ln(e) \) played a crucial role in solving for the constants \( A \) and \( B \). When manipulating logarithms, always remember these properties for efficient problem-solving.

solving equations

Solving equations is a fundamental skill in mathematics, allowing us to find unknown values in expressions and statements. In the exercise provided, we face a system of equations which is a set of two or more equations with the same variables.

Here is how you solve a simple system of equations step by step:

• First, understand the problem and identify what you need to find. For the problem at hand, our goal was to find the constants \( A \) and \( B \) in the function \( f(x) = A \ln(x) + B \).
• Next, substitute the given values into the equations provided. This gave us two equations: \( 10 = 0 + B \) and \( 1 = A + B \).
• Then, simplify the equations by solving for one variable in terms of the other. From \( 10 = B \), we easily found \( B = 10 \).
• Finally, substitute back into the second equation to find the other variable: \( 1 = A + 10 \), leading us to \( A = -9 \).

The key is to systematically simplify and substitute until all unknowns are determined. This systematic approach can be widely applied to different kinds of linear systems of equations.

constants in functions

In mathematical functions, constants refer to fixed values that do not change. In our exercise, \( A \) and \( B \) are constants in the function \( f(x) = A \ln(x) + B \). Understanding the role of constants can help in writing and transforming functions effectively.

Here's how constants function in equations:

• Constants determine specific characteristics of the function graph, influencing its position and shape. For instance, \( B \) as a constant shift moves the entire graph of the logarithmic function up or down depending on its value.
• In the provided function, \( A \) serves as the coefficient of the natural logarithm, affecting the steepness of the curve. A negative \( A \), like \( -9 \), reflects the graph about the x-axis.
• Using given conditions, like \( f(1) = 10 \) and \( f(e) = 1 \), we solved to find the constants. These conditions can be thought of as anchors, fixing the function's behavior at specific points.

Recognizing and identifying constants in functions is critical in defining and interpreting the behavior of mathematical models across various real-world contexts.